Hi Jennifer,
Since my experiences have been quite different from some others', thought some comments might be in order. My conclusions are consistent with Roger's at least, so I don't feel like an outlier.
A most irritating feature of low denier (10D-30D) nylon, is that like an octopus or a jellyfish, it really has no set dimensions. For this reason, for longer seams I use fine pins to connect the fabric pieces where the seam will be, and when marking the fabric for cutting, make sure the tension is just enough to hold it flat on the cutting table, and no more.
But that is just the beginning. Experiments must be done with samples of the material and the intended thread at different tensions to make sure beforehand that a flat, unpuckered seam will result.
Then the trick is to hold the work at both ends, or both in front and in back of the needle, while the work is being sewn. This requires holding the work with just the right amount of tension, not too much, not too little, to keep it flat, while maintaining the shoulder joints as loose as possible so they don't interfere with the passage of the material under the presser foot. Sort of like paddling a canoe or kayak: the shaft is gripped tightly and the elbows kept straight, but the shoulders are loose. Some recommend some of the varieties of rolling feet for this, but KISS is my guiding principle.
Also, the thread tensions, both top and bottom, must be set low enough so that the thread does not cause puckering. And the thread should not be too heavy for such a gossamer fabric. In the USA, Mettler Metrosene Plus is a light outdoor thread available in many colors, a polyester covered polyester, found at many outlets, although not the chains. Good strength to weight.
The pins are removed just before the work passes under the needle. They are there just to make sure the fabrics are joined at the points where you want them to be joined.
In this fashion, a seam without puckering is obtained, and the seam ends where it was marked to end on all pieces being sewn.
Often the so-called DWR (durable water resistant) coating on nylons is just a thinner coating of silicone that presents the same resistance to adhesives that the thicker coatings present on so-called silnylons. With the pins, this is a non-issue.
On a more general note, sewing nylons is a little like hiking poorly marked trails. (The Forest Service calls them 'undefined,'or something to that effect. Translation: We're too lazy to get out and mark the trails.) When in doubt, do not 'soldier on.' Back up to where you knew where you were going. With sewing, that often means removing seams with a sharp seam ripper. Less work in the long run.
These approaches have worked for me time and time again without fail, so thought they deserved mention. Best of luck.